Directed Polymers to the Critical SHF via moments

The Critical Stochastic Heat Flow (SHF) was originally constructed by Caravenna, Sun, and Zygouras (2023), as the scaling limit of partition functions of two-dimensional directed polymers in the critical disorder regime. Its moments are closely related to the two-dimensional delta Bose gas. I will discuss this connection in some detail, including the explicit description of the delta Bose semigroup obtained by Gu, Quastel, and Tsai (2019) through a resolvent expansion.

Recently, Tsai (2024) gave an axiomatic characterization of the critical SHF based on its moments together with a few additional properties. I will briefly explain the main ideas behind the proof of this characterization. As an application, I will also discuss a recent result proving convergence of partition functions to the critical SHF for a class of polymers in spatially correlated Brownian environments.